arXiv subjects

Uzy Smilansky

Publications and source records attributed to Uzy Smilansky.

At least 37 records · Page 2Linked to original sources

arXiv2014

Transmission through a noisy network

Quantum graphs with leads to infinity serve as convenient models for studying various aspects of systems which are usually attributed to chaotic scattering. They are also studied in several experimental systems and practical applications. In the present manuscript we investigate the effect of a time dependent random noise on the transmission of such graphs, and in particular on the resonances which dominate the scattering observable such as e.g., the transmission and reflection intensities. We model the noise by a potential $αδ(x-(x_0 +γ(t)))$ localized at an arbitrary point $x_0$ on any of the graph bonds, that fluctuates in time as a Brownian particle bounded in a harmonic potential described by the Ornstein-Uhlenbeck statistics. This statistics, which binds the Brownian motion within a finite interval, enables the use of a second order time-dependent perturbation theory, which can be applied whenever the strength parameter $α$ is sufficiently small. The theoretical frame-work will be explained in full generality, and will be explicitly solved for a simple, yet nontrivial example.

nlin.CD↗

arXiv2012

Topological Resonances in Scattering on Networks (Graphs)

We report on a hitherto unnoticed type of resonances occurring in scattering from networks (quantum graphs) which are due to the complex connectivity of the graph - its topology. We consider generic open graphs and show that any cycle leads to narrow resonances which do not fit in any of the prominent paradigms for narrow resonances (classical barriers, localization due to disorder, chaotic scattering). We call these resonances `topological' to emphasize their origin in the non-trivial connectivity. Topological resonances have a clear and unique signature which is apparent in the statistics of the resonance parameters (such as e.g., the width, the delay time or the wave-function intensity in the graph). We discuss this phenomenon by providing analytical arguments supported by numerical simulation, and identify the features of the above distributions which depend on genuine topological quantities such as the length of the shortest cycle (girth). These signatures cannot be explained using any of the other paradigms for narrow resonances. Finally, we propose an experimental setting where the topological resonances could be demonstrated, and study the stability of the relevant distribution functions to moderate dissipation.

nlin.CD↗

arXiv2012

Trace Formulae for quantum graphs with edge potentials

This work explores the spectra of quantum graphs where the Schrödinger operator on the edges is equipped with a potential. The scattering approach, which was originally introduced for the potential free case, is extended to this case and used to derive a secular function whose zeros coincide with the eigenvalue spectrum. Exact trace formulas for both smooth and $δ$-potentials are derived, and an asymptotic semiclassical trace formula (for smooth potentials) is presented and discussed.

math-ph↗

arXiv2012

Critical partitions and nodal deficiency of billiard eigenfunctions

The paper addresses the the number of nodal domains for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a "nodal deficiency" arises. Examples are known of eigenfunctions with arbitrarily large index $n$ that have just two nodal domains. It was suggested in the recent years to look at the partitions of the domain, rather than eigenfunctions. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) bipartite partitions minimizing the maximum of the ground-state energies in sub-domains of the partition, correspond to the "Courant sharp" eigenfunctions, i.e. to those with zero nodal deficiency. In this paper, the authors show, under some genericity conditions, among the bipartite equipartitions, the nodal ones correspond exactly to the critical points of an analogous functional, with the nodal deficiency being equal to the Morse index at this point. This explains, in particular, why all the minimal partitions must be Courant sharp.

math-ph↗

arXiv2011

Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by $ν_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-ν_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.) The main result of the present work is that the nodal deficiency for generic eigenvectors equals to a Morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue. The association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs [arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$ [arXiv:1107.3489]. The present work adapts this result to the discrete case. The definition of the energy functional in the discrete case requires a special setting, substantially different from the one used in [arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.

math-ph↗

arXiv2011

Note on the role of symmetry in scattering from isospectral graphs and drums

We discuss scattering from pairs of isospectral quantum graphs constructed using the method described in [1, 2]. It was shown in [3] that scattering matrices of such graphs have the same spectrum and polar structure, provided that infinite leads are attached in a way which preserves the symmetry of isospectral construction. In the current paper we compare this result with the conjecture put forward by Okada et al. [4] that the pole distribution of scattering matrices in the exterior of isospectral domains in R^2 are different.

math-ph↗

arXiv2011

Periodic Walks on Large Regular Graphs and Random Matrix Theory

We study the distribution of the number of (non-backtracking) periodic walks on large regular graphs. We propose a formula for the ratio between the variance of the number of $t$-periodic walks and its mean, when the cardinality of the vertex set $V$ and the period $t$ approach $\infty$ with $t/V\rightarrow τ$ for any $τ$. This formula is based on the conjecture that the spectral statistics of the adjacency eigenvalues is given by Random Matrix Theory (RMT). We provide numerical and theoretical evidence for the validity of this conjecture. The key tool used in this study is a trace formula which expresses the spectral density of $d$-regular graphs, in terms of periodic walks.

math-ph↗

arXiv2011

On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $ν_n$ of the $n$-th eigenfunction satisfies $n\ge ν_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-ν_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

math-ph↗

arXiv2011

Stationary scattering from a nonlinear network

Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. We show that the existence of cycles inside the graph leads to a large number of sharp resonances that dominate scattering. The latter resonances are then shown to be extremely sensitive to the nonlinearity and display multi-stability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks.

nlin.CD↗

arXiv2010

Scattering from isospectral quantum graphs

Quantum graphs can be extended to scattering systems when they are connected by leads to infinity. It is shown that for certain extensions, the scattering matrices of isospectral graphs are conjugate to each other and their poles distributions are therefore identical. The scattering matrices are studied using a recently developed isospectral theory. At the same time, the scattering approach offers a new insight on the mentioned isospectral construction.

math-ph↗

arXiv2010

Dynamics of nodal points and the nodal count on a family of quantum graphs

We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of view of the dynamics of zeros of the solutions to the scattering problem.

math-ph↗

arXiv2010

Trace formula for counting nodal domains on the boundaries of chaotic 2D billiards

Given a Dirichlet eigenfunction of a 2D quantum billiard, the boundary domain count is the number of intersections of the nodal lines with the boundary. We study the integer sequence defined by these numbers, sorted according to the energies of the eigenfunctions. Based on a variant of Berry's random wave model, we derive a semi-classical trace formula for the sequence of boundary domain counts. The formula consists of a Weyl-like smooth part, and an oscillating part which depends on classical periodic orbits and their geometry. The predictions of this trace formula are supported by numerical data computed for the Africa billiard.

quant-ph↗

arXiv2010

Level sets percolation on chaotic graphs

One of the most surprising discoveries in quantum chaos was that nodal domains of eigenfunctions of quantum-chaotic billiards and maps in the semi-classical limit display critical percolation. Here we extend these studies to the level sets of the adjacency eigenvectors of d-regular graphs. Numerical computations show that the statistics of the largest level sets (the maximal connected components of the graph for which the eigenvector exceeds a prescribed value) depend critically on the level. The critical level is a function of the eigenvalue and the degree d. To explain the observed behavior we study a random Gaussian waves ensemble over the d-regular tree. For this model, we prove the existence of a critical threshold. Using the local tree property of d-regular graphs, and assuming the (local) applicability of the random waves model, we can compute the critical percolation level and reproduce the numerical simulations. These results support the random-waves model for random regular graphs and provides an extension to Bogomolny's percolation model for two-dimensional chaotic billiards.

math-ph↗

arXiv2010

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (II)

Following the derivation of the trace formulae in the first paper in this series, we establish here a connection between the spectral statistics of random regular graphs and the predictions of Random Matrix Theory (RMT). This follows from the known Poisson distribution of cycle counts in regular graphs, in the limit that the cycle periods are kept constant and the number of vertices increases indefinitely. The result is analogous to the so called "diagonal approximation" in Quantum Chaos. We also show that by assuming that the spectral correlations are given by RMT to all orders, we can compute the leading deviations from the Poisson distribution for cycle counts. We provide numerical evidence which supports this conjecture.

math-ph↗

arXiv2009

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.

math-ph↗

arXiv2008

Exterior-Interior Duality for Discrete Graphs

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in $\mathbb{R}^2$, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Interior duality is derived for both methods.

math-ph↗

arXiv2008

Nodal domains on graphs - How to count them and why?

The purpose of the present manuscript is to collect known results and present some new ones relating to nodal domains on graphs, with special emphasize on nodal counts. Several methods for counting nodal domains will be presented, and their relevance as a tool in spectral analysis will be discussed.

math-ph↗

arXiv2008

Counting nodal domains on surfaces of revolution

We consider eigenfunctions of the Laplace-Beltrami operator on special surfaces of revolution. For this separable system, the nodal domains of the (real) eigenfunctions form a checker-board pattern, and their number $ν_n$ is proportional to the product of the angular and the "surface" quantum numbers. Arranging the wave functions by increasing values of the Laplace-Beltrami spectrum, we obtain the nodal sequence, whose statistical properties we study. In particular we investigate the distribution of the normalized counts $\frac{ν_n}{n}$ for sequences of eigenfunctions with $K \le n\le K + ΔK$ where $K,ΔK \in \mathbb{N}$. We show that the distribution approaches a limit as $K,ΔK\to\infty$ (the classical limit), and study the leading corrections in the semi-classical limit. With this information, we derive the central result of this work: the nodal sequence of a mirror-symmetric surface is sufficient to uniquely determine its shape (modulo scaling).

nlin.CD↗